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Cell Surface Tessellation: Abstract Context: Tumors of cuboidal or columnar epithelium are among the most common human malignancies. In benign cuboidal or columnar epithelium, the cell surface exhibits a regular, repeated packing of cells, resembling a collection of equal cylinders resting side-by-side. Malignant transformation involves the apparently independent features of variably-sized cells, variable nuclear ploidy, a disorganized surface, and tendency to invade surrounding tissues. Technology: Mathematically, a TILING is a plane-filling arrangement of plane figures, or its generalization to higher dimensions; a TESSELLATION is a periodic tiling of the plane by polygons, or space by polyhedra. Design: The cell surface is a tessellation of nearly-circular cell-apices. Each cell-pair has a unique tangent-line passing through a unique tangent-point; and each cell-triple has a unique line- segment drawn from the center of one cell to the opposite tangent-point. A cell-triple is BALANCED if and only if these six lines meet at a single intersection point. Results: It is demonstrated that a cell-triple is balanced if and only if all three cell-radii are equal. Conclusion: Malignant surface cells are characterized by more size variation and less balanced packing. In this model, unequal cell size and cell disorientation are geometric features of the same underlying process. Therapy for one process might possibly control the other process. Mathematical models can be used to propose alternatives to classical hypotheses in pathology, and explore general paradigms.

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Mathematical Tessellation 1. Tiling: plane-filling arrangement of plane figures, or generalization to higher dimensions. 2. Mathematically: tiling is a collection of disjoint open sets, the closures of which cover the plane. 3. Tessellation: periodic tiling of the plane by polygons, or space by polyhedra. 4. Seen in many drawings by M. C. Escher.

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Tessellation

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Cross-section: Picket Fence

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En-face: Honeycomb

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En-face: Malignancy

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Nearly-Circular Cell Apices

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Cell Surface Tessellation Nearly-circular cell-apices. Each cell-pair has a unique TANGENT-LINE passing through a unique tangent-point. Each cell-triple has a unique CENTER- OPPOSITE-LINE drawn from center of cell to the opposite tangent-point. Cell-triple is BALANCED if and only if these six lines meet at a single intersection point.

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Tangent-line. Center-opposite-line.

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Balanced/Unbalanced Cell Triples

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Mutually Tangent Circle Theorem Tangent-lines and Center- opposite-lines intersect at a common point if and only if all three cell-radii are equal. Proof of If: High-school geometry. Circles, radius=1; all six points lie at coordinates: (0, 1/√3). Proof of Only-If: Advanced problem.

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Summary: Mutually Tangent Circle Theorem Tangent-lines and center-opposite-lines intersect at a common point if and only if all three cell-radii are equal.

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Struble Triangle Theorem (i). There exists a unique interior point D, for which the three line segments emanating from the vertices and passing through D, intersect the edges of the triangle at three opposing points, a, b and c, satisfying length equalities Ab=Ac, Ba=Bc and Ca=Cb. (ii). There exists a unique interior point E, for which three line segments emanating from E to the points a, b and c are perpendicular to the edges of the triangle.

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Struble Triangle Theorem (iii). There exists a unique interior point F and positive number r, for which three line segments emanating from the vertices to F have lengths, when shortened by r, given by Ab, Bc and Ca. (iv). The interior points D, E and F are coincident only for equilateral triangles.

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Mathematical Theories Can be used as alternatives to conventional models in pathology. Conventional model of cancer: invasion after tumor cells break through basement membrane. Alternative model of cancer: tumor proliferation as a property of cells, attempting to balance with neighboring cells.

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Possible Implications for Therapy Common-intersection and equal-radii properties equivalent. Processes are mathematically equivalent. Control one process, then you can control the other.

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Summary 5. Cell-triple radii are equal if and only if six lines meet at one point. 6. Cell disorientation and radius-equality are geometric features of same process. 7. Therapy for one process might possibly control the other process. 8. Mathematical models can be used to propose alternatives to classical hypotheses in pathology.